Local unitary representations of the braid group and their applications to quantum computing

2016/04/21 by Colleen Delaney, Eric C. Rowell, Delaney, Colleen +3 · 1 voice · 4 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Alexander polynomial #Algebra over a field #Algebraic structures and combinatorial models #Algorithm #Black Holes and Theoretical Physics #Braid #Braid group #Braid theory #Computation #Group (periodic table) #Knot theory #Lie group #Mathematics #Physics #Polynomial #Pure mathematics #Quantum #Quantum computer #Quantum mechanics #Representation (politics) #Symmetric group #Topological quantum computer #Unitary group #Unitary representation #Unitary state #math.QA

paper · pdf · doi:10.48550/arxiv.1604.06429

published in Repositorio Institucional UN - Biblioteca Digital

arxiv created 2016/04/21 · openalex publication_date 2016/04/21 · arxiv updated 2016/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We provide an elementary introduction to topological quantum computation based on the Jones representation of the braid group. We first cover the Burau representation and Alexander polynomial. Then we discuss the Jones representation and Jones polynomial and their application to anyonic quantum computation. Finally we outline the approximation of the Jones polynomial and explicit localizations of braid group representations.

Cited by

• Fusion Trees and Homological Representations
• Eventually periodic resolutions with applications to integral group rings
• On the irreducibility of singular braid group representations
• Hecke algebra representations from the Katz-Long-Moody construction
• On extensions of the standard representation of the braid group to the singular braid group

Discussions

• Theoretician explains experimentalist how to construct a universal quantum computer. arxiv.org/abs/1604.06429 [bsky, 21 points, 2 comments]

Related

• Quantum Computing and the Jones Polynomial
• The Jones polynomial: quantum algorithms and applications in quantum complexity theory
• Quantum Computation of Jones' Polynomials
• q - Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation
• Braids, Motions and Topological Quantum Computing